Discrete Theory of Gravity as a local Theory of the Poincare Group in the
First Order Formalism

Abstract:

We propose a discrete theory of gravity locally invariant under the Poincare group. We define a first order theory, in the sense of Palatini, of Regge Calculus on the metric-dual Voronoi complex of the original simplicial Regge complex in the same spirit of the continuum theory of General Relativity in Cartan formalism. The action results to be Wilson-like as in Lattice Gauge Theory. We show that the field equation can be carefully derived taking in account the constraints of the theory and they look very similar to the first order Einstein equations in the Cartan formalism. We will show that in the limit of small deficit angles these equations have Regge Calculus, locally, as the only solution. A quantum measure is easly defined which does not suffer the ambiguities of Regge Calculus, and a coupling with fermionic matter is easily introduced.